Finite Group Databasebeta

← #25068966

Subgroups — #25068966

Order 11008.

1654 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583461C211yesmaximal, characteristic, complemented
2#19143847C212yesmaximal, complemented
3#19133546C214yesmaximal, complemented
4#19133512C214yesmaximal, complemented
5#19133123C212yesmaximal, complemented
6#19132922C211yesmaximal, characteristic, complemented
7C43×#2404C211yesmaximal, characteristic, complemented
8C23×C22⋊D43—21no—
9C2×#13063350—22no—
10#14583461—24no—
11#14583461—22no—
12#14583461C2214yes—
13C2×#13063309—22no—
14C2×#13063299—21no—
15C2×#13063299C2212yes—
16C2×#13063295—22no—
17#14583044—24no—
18#14583038C2214yescomplemented
19#14583033C2214yescomplemented
20#14582900—22no—
21#14582863C2212yescomplemented
22#14582632C2218yescomplemented
23#14582623C2214yescomplemented
24C22×#41×C43C2211yescharacteristic, complemented
25C2×#120×C43—21no—
26C22×C22⋊C4×C43C2211yescharacteristic
27C43×#393C2212yescomplemented
28C43×#392C2212yescomplemented
29C43×#386—21no—
30C43×#386—22no—
31C43×#381C2211yescharacteristic, complemented
32C24×D43—21no—
33C22×D172—21no—
34C22×D172D412yescomplemented
35C22×C22⋊D43—22no—
36C22×C22⋊D43—24no—
37C22×C22⋊D43—22no—
38C22×C22⋊D43—24no—
39C22×C22⋊D43—48no—
40C23×Dic43—21no—
41#13063350—48no—
42C2×#10621884—24no—
43C2×#10621882—24no—
44C2×#10621877—22no—
45#13063309—28no—
46#13063309—28no—
47#13063299—24no—
48#13063299—24no—
49#13063299—22no—
50#13063299—22no—
51#13063299D414yescomplemented
52#13063299C2314yes—
53#13063295—28no—
54#13063295—28no—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43D412yescomplemented
57C2×(C2×D43)⋊C4—22no—
58C2×(C2×D43)⋊C4—22no—
59#13063249—24no—
60#13063246—24no—
61C22⋊(C43⋊C8)—22no—
62C22⋊(C43⋊C8)C2314yescomplemented
63C25×C43D411yescharacteristic
64C2×#41×C43—22no—
65C2×#41×C43—22no—
66C2×#41×C43—44no—
67C22×Q8×C43C2311yescharacteristic, complemented
68C22×D4×C43—21no—
69C22×D4×C43—22no—
70C22×D4×C43C2311yescharacteristic, complemented
71C23×C4×C43—21no—
72C23×C4×C43D411yescharacteristic
73C23×C4×C43C2311yescharacteristic
74#120×C43—44no—
75C2×C22⋊C4×C43—22no—
76C2×C22⋊C4×C43—22no—
77C2×C22⋊C4×C43—21no—
78C2×C22⋊C4×C43—22no—
79C2×C22⋊C4×C43—22no—
80C2×C22⋊C4×C43D412yes—
81C2×C22⋊C4×C43C2312yes—
82#95×C43—21no—
83#95×C43—22no—
84#95×C43C2312yescomplemented
85C24×C43—21no—
86C24×C43—21no—
87C24×C43—24no—
88C24×C43—21no—
89C24×C43—42no—
90C24×C43—81no—
91C24×C43C2×D411yescharacteristic
92C24×C43C2×D412yes—
93C23×D43—22no—
94C23×D43—24no—
95C23×D43—44no—
96C23×D43—82no—
97C23×D43—82no—
98#41×C43—28no—
99#41×C43—44no—
100#41×C43—44no—
101#41×C43—44no—
102C2×Q8×C43—22no—
103C2×Q8×C43—42no—
104C2×D4×C43—24no—
105C2×D4×C43—22no—
106C2×D4×C43—42no—
107C2×D4×C43—42no—
108C2×D4×C43—44no—
109C2×D4×C43—41no—
110C2×D4×C43—41no—
111C2×D4×C43—44no—
112C2×D4×C43—82no—
113C22×C4×C43—21no—
114C22×C4×C43—21no—
115C22×C4×C43—21no—
116C22×C4×C43—22no—
117C22×C4×C43—42no—
118C22×C4×C43—42no—
119C22×C4×C43—42no—
120C22×C4×C43—42no—
121C22×C4×C43—81no—
122C22×C4×C43C2×D412yes—
123C22×C4×C43C2×D412yes—
124C22×C4×C43C2×D412yes—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2411yescharacteristic
127C2×C22⋊D43—28no—
128C2×C22⋊D43—24no—
129C2×C22⋊D43—24no—
130C2×C22⋊D43—48no—
131C2×C22⋊D43—416no—
132C2×C22⋊D43—48no—
133C2×C22⋊D43—48no—
134C2×C22⋊D43—48no—
135C22×Dic43—24no—
136C22×Dic43—22no—
137C22×Dic43—44no—
138C2×D172—24no—
139C2×D172—22no—
140C2×D172—44no—
141C2×D172—82no—
142C2×D172C2×D414yes—
143C22⋊C4×C43—24no—
144C22⋊C4×C43—24no—
145C22⋊C4×C43—42no—
146C22⋊C4×C43—44no—
147C22⋊C4×C43—82no—
148#10621884—48no—
149#10621882—48no—
150(C2×D43)⋊C4—28no—
151(C2×D43)⋊C4—24no—
152(C2×D43)⋊C4—24no—
153C4⋊Dic43—24no—
154C4⋊Dic43—22no—
155C4⋊Dic43C2×D414yes—
156#10621877—44no—
157C2×C43⋊C8—24no—
158C23×C43—24no—
159C23×C43—24no—
160C23×C43—22no—
161C23×C43—22no—
162C23×C43—22no—
163C23×C43—22no—
164C23×C43—22no—
165C23×C43—41no—
166C23×C43—42no—
167C23×C43—42no—
168C23×C43—42no—
169C23×C43—42no—
170C23×C43—41no—
171C23×C43—42no—
172C23×C43—41no—
173C23×C43—42no—
174C23×C43—81no—
175C23×C43—81no—
176C23×C43—88no—
177C23×C43#12011yescharacteristic
178C23×C43#12011yescharacteristic
179C23×C43AGL(1,ℤ/8)14yescomplemented
180C23×C43C22×D411yescharacteristic
181C22×D43—24no—
182C22×D43—44no—
183C22×D43—44no—
184C22×D43—48no—
185C22×D43—44no—
186C22×D43—84no—
187C22×D43—84no—
188C22×D43—816no—
189Q8×C43—24no—
190Q8×C43—42no—
191D4×C43—24no—
192D4×C43—42no—
193D4×C43—44no—
194D4×C43—44no—
195D4×C43—88no—
196C2×C4×C43—24no—
197C2×C4×C43—22no—
198C2×C4×C43—22no—
199C2×C4×C43—41no—
200C2×C4×C43—42no—
201C2×C4×C43—41no—
202C2×C4×C43—48no—
203C2×C4×C43—42no—
204C2×C4×C43—41no—
205C2×C4×C43—41no—
206C2×C4×C43—42no—
207C2×C4×C43—42no—
208C2×C4×C43—42no—
209C2×C4×C43—42no—
210C2×C4×C43—81no—
211C2×C4×C43#12012yes—
212C2×C4×C43C22×D412yes—
213C22⋊D43—416no—
214C22⋊D43—48no—
215C22⋊D43—48no—
216C2×Dic43—24no—
217C2×Dic43—44no—
218C2×Dic43—48no—
219D172—44no—
220D172—82no—
221D172—84no—
222C43⋊C8—44no—
223#18132—431nomaximal, Sylow
224C22×C43—21no—
225C22×C43—21no—
226C22×C43—21no—
227C22×C43—21no—
228C22×C43—24no—
229C22×C43—21no—
230C22×C43—21no—
231C22×C43—21no—
232C22×C43—44no—
233C22×C43—44no—
234C22×C43—48no—
235C22×C43—44no—
236C22×C43—42no—
237C22×C43—42no—
238C22×C43—42no—
239C22×C43—82no—
240C22×C43—82no—
241C22×C43—84no—
242C22×C43—81no—
243C22×C43—81no—
244C22×C43B(4,𝔽2)12yes—
245C22×C43C2×#12011yescharacteristic
246C22×C43C2×AGL(1,ℤ/8)12yes—
247C2×D43—44no—
248C2×D43—44no—
249C2×D43—44no—
250C2×D43—88no—
251C2×D43—816no—
252C4×C43—22no—
253C4×C43—22no—
254C4×C43—42no—
255C4×C43—42no—
256C4×C43—82no—
257Dic43—44no—
258C2×#534—431no—
259#3006—432no—
260#2404D4311yescharacteristic, complemented
261#1601—434no—
262#1595—434no—
263#1510—432no—
264#1462—431no—
265C2×C43—22no—
266C2×C43—22no—
267C2×C43—44no—
268C2×C43—41no—
269C2×C43—82no—
270C2×C43#300311yescharacteristic
271C2×C43#301911yescharacteristic
272C2×C43C2×B(4,𝔽2)11yescharacteristic
273D43—84no—
274C22×#41C2×D4311yescharacteristic, complemented
275C23×D4—861no—
276C2×AGL(1,ℤ/8)—862no—
277#534—434no—
278#534—864no—
279#534—862no—
280C2×#123—862no—
281C2×#121—432no—
282C2×#121—861no—
283C2×#120—21no—
284C2×#120—862no—
285C22×C22⋊C4C2×D4311yescharacteristic
286#449—864no—
287#448—434no—
288#447—434no—
289#417—862no—
290#407—432no—
291#393C2×D4312yescomplemented
292#392C2×D4312yescomplemented
293#386—21no—
294#386—22no—
295#381C2×D4311yescharacteristic, complemented
296#327—438no—
297#323—434no—
298C43#1813211yescharacteristic, complemented, Sylow
299C25—861no—
300C25C22⋊D4311yescharacteristic
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—44no—
304C22×Q8C22×D4311yescharacteristic, complemented
305C22×D4—21no—
306C22×D4—22no—
307C22×D4—432no—
308C22×D4—861no—
309C22×D4—862no—
310C22×D4—864no—
311C22×D4—862no—
312C22×D4—864no—
313C22×D4—1728no—
314C22×D4C22×D4311yescharacteristic, complemented
315C23×C4—21no—
316C23×C4—861no—
317C23×C4C22⋊D4311yescharacteristic
318C23×C4C22×D4311yescharacteristic
319AGL(1,ℤ/8)—1728no—
320C2×SD16—864no—
321C2×D8—864no—
322C2×M16—862no—
323#123—868no—
324#123—868no—
325#121—434no—
326#121—434no—
327#121—864no—
328#121—862no—
329#121—862no—
330#121—864no—
331#120—44no—
332#120—868no—
333#120—868no—
334C2×C4⋊C4—432no—
335C2×C4⋊C4—861no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—862no—
342C2×C22⋊C4—862no—
343C2×C22⋊C4C22⋊D4312yes—
344C2×C22⋊C4C22×D4312yes—
345#103—864no—
346#102—864no—
347C22⋊C8—434no—
348C22⋊C8—862no—
349#95—21no—
350#95—22no—
351#95C22×D4312yescomplemented
352C24—21no—
353C24—21no—
354C24—24no—
355C24—21no—
356C24—42no—
357C24—81no—
358C24—864no—
359C24—862no—
360C24—3442no—
361C24—3442no—
362C24—1724no—
363C24C2×C22⋊D4311yescharacteristic
364C24C2×C22⋊D4312yes—
365#41—28no—
366#41—44no—
367#41—44no—
368#41—44no—
369C2×Q8—22no—
370C2×Q8—42no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—82no—
380C2×D4—434no—
381C2×D4—864no—
382C2×D4—868no—
383C2×D4—864no—
384C2×D4—864no—
385C2×D4—862no—
386C2×D4—3442no—
387C2×D4—1724no—
388C2×D4—1728no—
389C2×D4—17216no—
390C2×D4—1728no—
391C2×D4—1728no—
392C2×D4—1728no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—81no—
402C22×C4—864no—
403C22×C4—862no—
404C22×C4—1724no—
405C22×C4D4×D4312yes—
406C22×C4D4×D4312yes—
407C22×C4C2×C22⋊D4312yes—
408C22×C4C2×C22⋊D4311yescharacteristic
409C22×C4C23×D4311yescharacteristic
410SD16—1728no—
411D8—1728no—
412M16—1724no—
413C2×C8—864no—
414C4⋊C4—434no—
415C4⋊C4—864no—
416C4⋊C4—862no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—82no—
422C22⋊C4—868no—
423C22⋊C4—864no—
424C22⋊C4—864no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—81no—
442C23—81no—
443C23—88no—
444C23—864no—
445C23—34416no—
446C23—3444no—
447C23—3444no—
448C23—1724no—
449C23—1724no—
450C23—1728no—
451C23—1724no—
452C23#1306329311yescharacteristic
453C23#1306329311yescharacteristic
454C23#1306335014yescomplemented
455C23C22×C22⋊D4311yescharacteristic
456Q8—24no—
457Q8—42no—
458D4—24no—
459D4—42no—
460D4—44no—
461D4—44no—
462D4—88no—
463D4—3442no—
464D4—3444no—
465D4—1724no—
466D4—17216no—
467D4—1728no—
468D4—1728no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—864no—
485C2×C4—1724no—
486C2×C4—1728no—
487C2×C4#1306329312yes—
488C2×C4C2×D4×D4312yes—
489C8—1724no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—24no—
497C22—21no—
498C22—44no—
499C22—44no—
500C22—48no—
501C22—44no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—84no—
508C22—81no—
509C22—81no—
510C22—3448no—
511C22—34416no—
512C22—1724no—
513C22—1724no—
514C22—1724no—
515C22#1458308512yes—
516C22C2×#1306329311yescharacteristic
517C22C2×#1306335012yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—1724no—
524C2—22no—
525C2—22no—
526C2—41no—
527C2—44no—
528C2—82no—
529C2—3444no—
530C2#1914382111yescharacteristic
531C2#1914395111yescharacteristic
532C2C2×#1458308511yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2404C2
socleC22×C43C2×#120
hypercenter#2404D43
hypocenterC43#18132
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#14583461C2semidirect
2. #19143847C2semidirect
3. #19133546C2semidirect
4. #19133512C2semidirect
5. #19133123C2semidirect
6. #19132922C2semidirect
7. C43×#2404C2semidirect
18. #14583038C22semidirect
19. #14583033C22semidirect
21. #14582863C22semidirect
22. #14582632C22semidirect
23. #14582623C22semidirect
24. C22×#41×C43C22semidirect
27. C43×#393C22semidirect
28. C43×#392C22semidirect
31. C43×#381C22semidirect
34. C22×D172D4semidirect
51. #13063299D4semidirect
56. C2×C4⋊Dic43D4semidirect
62. C22⋊(C43⋊C8)C23semidirect
67. C22×Q8×C43C23semidirect
70. C22×D4×C43C23semidirect
84. #95×C43C23semidirect
179. C23×C43AGL(1,ℤ/8)semidirect
260. #2404D43semidirect
274. C22×#41C2×D43semidirect
291. #393C2×D43semidirect
292. #392C2×D43semidirect
295. #381C2×D43semidirect
298. C43#18132semidirect
304. C22×Q8C22×D43semidirect
314. C22×D4C22×D43semidirect
351. #95C22×D43semidirect
454. C23#13063350semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#145834612
#191438472
#191335462
#191335122
#191331232
#191329222
C43×#24042
#1813243